Solving Maxwell's equations can be either solved directly or by solving the wave equation.  Since the wave equation is derived from Maxwell's equations, any solution to the wave equation is also a solution to Maxwell's equations.  Another approach is to use vector potential theory and will be covered in a separate chapter.  

\section{Separation of Space and Time}
In this section, it will be shown how to use the \emph{separation of variables} technique to separate temporal variations from spatial variations.  This will lead to uncoupled differential equations that can be solved or further separated.
\subsection{Wave Equation}
\input{./src/SeparationOfSpaceAndTimeWaveEqn.tex}
\subsection{Temporal Solutions of Wave Equation}
\input{./src/temporalsolnwaveeqn.tex}
\subsection{Maxwell's Equations}
\input{./src/MaxwellseqnSeparationofTime.tex}


\section{Normal/Transverse Spatial Separation}
The focus of this section will be to use techniques to solve the spatial wave equation (\ref{eqn:waveeqnsoln}).  A certain class or family of solutions will be found by breaking the problem into \emph{normal} and \emph{transverse} components.  Coordinate systems that allow solutions under these conditions must have one direction that remains constant and will be classified as \emph{uniform} in contrast to all other solutions being  \emph{non-uniform} solutions.  Rectangular, cylindrical, and elliptical coordinate systems are examples of coordinate systems that meet these conditions.  The independent variables along with the operators that operate on them will also be separated into normal and transverse coordinates.
\subsection{Wave Equation}
\input{./src/waveeqtransversenormal.tex}\label{sec:waveeqtransversenormal}
\subsection{Traveling Wave Solutions}
\input{./src/travelingwaves.tex}\label{sec:travelingwaves}
\subsection{Standing Wave Solutions}
\input{./src/standingwaves.tex}\label{sec:standingwaves}
\subsection{TE and TM Mode Solutions}
\input{./src/TETMSoln.tex}

%\section{Rectangular Coordinate System}
%\input{./src/waveeqnrec.tex}
%\section{Cylindrical Coordinate System}
%\input{waveeqncyl.tex}

%\section{Temporal Solutions}
%\input{./src/SolnWaveEqn.tex}
%\section{Spherical Separation}
